A350132 - OEIS

%I #36 Sep 08 2024 19:55:35

%S 34,7,41,919,18089,446081,27033161,663929729,74335064959,6132592231039

%N a(n) is the smallest number m such that tau(m-1) = tau(m+1) = tau(m)^n, where tau(m) is the number of divisors of m (A000005).

%C Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n))^n, tau(a(n)), tau(a(n))^n]: [4, 4, 4], [4, 2, 4], [8, 2, 8], [16, 2, 16], [32, 2, 32], [64, 2, 64], [128, 2, 128], ...

%C Conjecture: a(n) is prime for all n >= 2, i.e., the sequence {a(n)} without the first term is the sequence of smallest primes p such that tau(p-1) = tau(p+1) = 2^n for n >= 2.

%C a(10) <= 6132592231039. - _Jon E. Schoenfield_, Jan 19 2022

%C From _David A. Corneth_, Jan 21 2022: (Start)

%C a(11) <= 864808145605249.

%C a(12) <= 246846832951283839.

%C a(13) <= 14552217960448488319. (End)

%e 34 is the 1st term of A169834, so a(1) = 34.

%o (Magma) Ax:=func<n|exists(r){m: m in[2..10^6] | #Divisors(m - 1) eq #Divisors(m) ^ n and #Divisors(m + 1) eq #Divisors(m) ^ n} select r else 0>; [Ax(n): n in [1..6]];

%o (PARI) isok(m, n) = my(nd1=numdiv(m-1)); (nd1 == numdiv(m)^n) && (nd1 == numdiv(m+1));

%o a(n) = {my(m=2); while (!isok(m, n), m++); m;} \\ _Michel Marcus_, Dec 16 2021

%Y Cf. A000005, A169834, A343020, A343087.

%K nonn,more

%O 1,1

%A _Jaroslav Krizek_, Dec 15 2021

%E a(8) from _Jon E. Schoenfield_ and _David A. Corneth_, Dec 15 2021

%E a(9) from _David A. Corneth_ and _Martin Ehrenstein_, Jan 14 2022

%E a(10) verified by _Martin Ehrenstein_, Jan 21 2022